36 research outputs found
Unavoidable Parallel Minors of 4-Connected Graphs
A parallel minor is obtained from a graph by any sequence of edge
contractions and parallel edge deletions. We prove that, for any positive
integer k, every internally 4-connected graph of sufficiently high order
contains a parallel minor isomorphic to a variation of K_{4,k} with a complete
graph on the vertices of degree k, the k-partition triple fan with a complete
graph on the vertices of degree k, the k-spoke double wheel, the k-spoke double
wheel with axle, the (2k+1)-rung Mobius zigzag ladder, the (2k)-rung zigzag
ladder, or K_k. We also find the unavoidable parallel minors of 1-, 2-, and
3-connected graphs.Comment: 12 pages, 3 figure
Large Non-Planar Graphs and an Application to Crossing-Critical Graphs
We prove that, for every positive integer k, there is an integer N such that
every 4-connected non-planar graph with at least N vertices has a minor
isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding
an edge joining every pair of vertices at distance exactly k, or the graph
obtained from a cycle of length k by adding two vertices adjacent to each other
and to every vertex on the cycle. We also prove a version of this for
subdivisions rather than minors, and relax the connectivity to allow 3-cuts
with one side planar and of bounded size. We deduce that for every integer k
there are only finitely many 3-connected 2-crossing-critical graphs with no
subdivision isomorphic to the graph obtained from a cycle of length 2k by
joining all pairs of diagonally opposite vertices.Comment: To appear in Journal of Combinatorial Theory B. 20 pages. No figures.
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On inequivalent representations of matroids over finite fields
Kahn conjectured in 1988 that, for each prime power q, there is an integer n(q) such that no 3-connected GF(q)-representable matroid has more than n(q) inequivalent GF(q)-representations. At the time, this conjecture was known to be true for q = 2 and q = 3, and Kahn had just proved it for q = 4. In this paper, we prove the conjecture for q = 5, showing that 6 is a sharp value for n(5). Moreover, we also show that the conjecture is false for all larger values of q. © 1996 Academic Press, Inc
A new approach to solving three combinatorial enumeration problems on planar graphs
The purpose of this paper is to show how the technique of delta-wye graph reduction provides an alternative method for solving three enumerative function evaluation problems on planar graphs. In particular, it is shown how to compute the number of spanning trees and perfect matchings, and how to evaluate energy in the Ising spin glass model of statistical mechanics. These alternative algorithms require O(n2) arithmetic operations on an n-vertex planar graph, and are relatively easy to implement
Totally free expansions of matroids
The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension M′ of M by an element x′ such that {x, x′} is independent and M′ is unaltered by swapping the labels on x and x′. When x is fixed, a representation of M.\x extends in at most one way to a representation of M. A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M, of the number of inequivalent F-representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r ≥ 4, there are unique and easily described rank-r quaternary and quinternary matroids, the first being the free spike. Finally, Seymour\u27s Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N, whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and cofixed elements occur in the initial segment of the sequence. © 2001 Elsevier Science
Unavoidable minors of large 3-connected matroids
This paper proves that, for every integernexceeding two, there is a numberN(n) such that every 3-connected matroid with at leastN(n) elements has a minor that is isomorphic to one of the following matroids: an (n+2)-point line or its dual, the cycle or cocycle matroid ofK3,n, the cycle matroid of a wheel withnspokes, a whirl of rankn, or ann-spike. A matroid is of the last type if it has ranknand consists ofnthree-point lines through a common point such that, for allkin {1,2,...,n-1}, the union of every set ofkof these lines has rankk+1. © 1997 Academic Press
Surfaces, Tree-Width, Clique-Minors, and Partitions
In 1971, Chartrand, Geller, and Hedetniemi conjectured that the edge set of a planar graph may be partitioned into two subsets, each of which induces an outerplanar graph. Some partial results towards this conjecture are presented. One such result, in which a planar graph may be thus edge partitioned into two series-parallel graphs, has nice generalizations for graphs embedded onto an arbitrary surface and graphs with no large clique-minor. Several open questions are raised. © 2000 Academic Press
Excluding any graph as a minor allows a low tree-width 2-coloring
This article proves the conjecture of Thomas that, for every graph G, there is an integer k such that every graph with no minor isomorphic to G has a 2-coloring of either its vertices or its edges where each color induces a graph of tree-width at most k. Some generalizations are also proved. © 2003 Elsevier Inc. All rights reserved